# First Principles Example 2: x³

In mathematical discourse, the concept of "first principles" serves as a foundational approach to understanding complex functions and equations. Exploring this notion through the lens of a specific example, consider the cubic function \( x^3 \). By dissecting this expression down to its fundamental elements, we unveil its inherent properties and behaviors without relying on preconceived notions or external assumptions. Analyzing \( x^3 \) from first principles offers a clear pathway to comprehend its fundamental characteristics, paving the way for deeper insights into its graphical representation, roots, and overall behavior across the real number line.

Questions

- What is the second derivative of f(x)=ln (x+3)?
- Find the antiderivative of f'(x)=3x^3?
- Use the definition of the derivative at a point to find an eq for the tangent line to y= x^3 at the point (1,1) . No points for any other methods help??
- For what value of #x# is the slope of the tangent line to #y=x^7 + (3/x)# undefined?
- How do you find the derivative of (-2)/(x(ln x)^3). ?
- Solve: dy/dx+ y = xy^3 .??
- If y = f(x)g(x), then dy/dx = f‘(x)g‘(x). If it is true, explain your answer. If false, provide a counterexample. True or False?
- Hi everyone! My question is: "find the general solution y(x) of the given differential equation": #xy'+y=x^6# How do I solve this? Thank you for your answer!
- This is the second of three derivative problems. Can someone please help solve it?
- Find the integral of #1/x^3+x#?
- What is the derivative of #f(x) = ln(x^3+3x^2)/(3x+6)#?
- The value of Lim x->1 (x^5-3x+2)/x-1 is equal to?
- What is the derivative of ln(1+x)?
- Integral of dx /x^2018 - x=?
- What is the Derivative of (x+(3˄x))˄4?
- What is the derivative of #f(x) = 1/(xlnx)#?
- Let #G(x)=int_x^{-2}cos(t^2)dt#. Then what is G'(x) ?
- Integral of #int(x-1)/(x+3)^3 dx# ?
- What is the derivative of f(x)=10(5.1)^x+21(0.22)^x+5e^x?
- What is the derivative of #y=x/(5-3x)#?